TSTP Solution File: NUM181-1 by Z3---4.8.9.0
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- Process Solution
%------------------------------------------------------------------------------
% File : Z3---4.8.9.0
% Problem : NUM181-1 : TPTP v8.1.0. Bugfixed v2.1.0.
% Transfm : none
% Format : tptp
% Command : z3_tptp -proof -model -t:%d -file:%s
% Computer : n006.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sun Sep 18 13:03:06 EDT 2022
% Result : Unsatisfiable 0.71s 0.73s
% Output : Proof 0.81s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.05/0.09 % Problem : NUM181-1 : TPTP v8.1.0. Bugfixed v2.1.0.
% 0.05/0.09 % Command : z3_tptp -proof -model -t:%d -file:%s
% 0.09/0.28 % Computer : n006.cluster.edu
% 0.09/0.28 % Model : x86_64 x86_64
% 0.09/0.28 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.09/0.28 % Memory : 8042.1875MB
% 0.09/0.28 % OS : Linux 3.10.0-693.el7.x86_64
% 0.09/0.28 % CPULimit : 300
% 0.09/0.28 % WCLimit : 300
% 0.09/0.28 % DateTime : Fri Sep 2 07:57:54 EDT 2022
% 0.09/0.28 % CPUTime :
% 0.09/0.28 Z3tptp [4.8.9.0] (c) 2006-20**. Microsoft Corp.
% 0.09/0.28 Usage: tptp [options] [-file:]file
% 0.09/0.28 -h, -? prints this message.
% 0.09/0.28 -smt2 print SMT-LIB2 benchmark.
% 0.09/0.28 -m, -model generate model.
% 0.09/0.28 -p, -proof generate proof.
% 0.09/0.28 -c, -core generate unsat core of named formulas.
% 0.09/0.28 -st, -statistics display statistics.
% 0.09/0.28 -t:timeout set timeout (in second).
% 0.09/0.28 -smt2status display status in smt2 format instead of SZS.
% 0.09/0.28 -check_status check the status produced by Z3 against annotation in benchmark.
% 0.09/0.28 -<param>:<value> configuration parameter and value.
% 0.09/0.28 -o:<output-file> file to place output in.
% 0.71/0.73 % SZS status Unsatisfiable
% 0.71/0.73 % SZS output start Proof
% 0.71/0.73 tff(subclass_type, type, (
% 0.71/0.73 subclass: ( $i * $i ) > $o)).
% 0.71/0.73 tff(universal_class_type, type, (
% 0.71/0.73 universal_class: $i)).
% 0.71/0.73 tff(intersection_type, type, (
% 0.71/0.73 intersection: ( $i * $i ) > $i)).
% 0.71/0.73 tff(ordinal_numbers_type, type, (
% 0.71/0.73 ordinal_numbers: $i)).
% 0.71/0.73 tff(complement_type, type, (
% 0.71/0.73 complement: $i > $i)).
% 0.71/0.73 tff(union_type, type, (
% 0.71/0.73 union: ( $i * $i ) > $i)).
% 0.71/0.73 tff(image_type, type, (
% 0.71/0.73 image: ( $i * $i ) > $i)).
% 0.71/0.73 tff(successor_relation_type, type, (
% 0.71/0.73 successor_relation: $i)).
% 0.71/0.73 tff(singleton_type, type, (
% 0.71/0.73 singleton: $i > $i)).
% 0.71/0.73 tff(null_class_type, type, (
% 0.71/0.73 null_class: $i)).
% 0.71/0.73 tff(member_type, type, (
% 0.71/0.73 member: ( $i * $i ) > $o)).
% 0.71/0.73 tff(limit_ordinals_type, type, (
% 0.71/0.73 limit_ordinals: $i)).
% 0.71/0.73 tff(unordered_pair_type, type, (
% 0.71/0.73 unordered_pair: ( $i * $i ) > $i)).
% 0.71/0.73 tff(1,assumption,(member(null_class, universal_class)), introduced(assumption)).
% 0.71/0.73 tff(2,plain,
% 0.71/0.73 (^[Y: $i, X: $i] : refl((complement(intersection(complement(X), complement(Y))) = union(X, Y)) <=> (complement(intersection(complement(X), complement(Y))) = union(X, Y)))),
% 0.71/0.73 inference(bind,[status(th)],[])).
% 0.71/0.73 tff(3,plain,
% 0.71/0.73 (![Y: $i, X: $i] : (complement(intersection(complement(X), complement(Y))) = union(X, Y)) <=> ![Y: $i, X: $i] : (complement(intersection(complement(X), complement(Y))) = union(X, Y))),
% 0.71/0.73 inference(quant_intro,[status(thm)],[2])).
% 0.71/0.73 tff(4,plain,
% 0.71/0.73 (![Y: $i, X: $i] : (complement(intersection(complement(X), complement(Y))) = union(X, Y)) <=> ![Y: $i, X: $i] : (complement(intersection(complement(X), complement(Y))) = union(X, Y))),
% 0.71/0.73 inference(rewrite,[status(thm)],[])).
% 0.71/0.73 tff(5,axiom,(![Y: $i, X: $i] : (complement(intersection(complement(X), complement(Y))) = union(X, Y))), file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax','union')).
% 0.71/0.73 tff(6,plain,
% 0.71/0.73 (![Y: $i, X: $i] : (complement(intersection(complement(X), complement(Y))) = union(X, Y))),
% 0.71/0.73 inference(modus_ponens,[status(thm)],[5, 4])).
% 0.71/0.73 tff(7,plain,(
% 0.71/0.73 ![Y: $i, X: $i] : (complement(intersection(complement(X), complement(Y))) = union(X, Y))),
% 0.71/0.73 inference(skolemize,[status(sab)],[6])).
% 0.71/0.73 tff(8,plain,
% 0.71/0.73 (![Y: $i, X: $i] : (complement(intersection(complement(X), complement(Y))) = union(X, Y))),
% 0.71/0.73 inference(modus_ponens,[status(thm)],[7, 3])).
% 0.71/0.73 tff(9,plain,
% 0.71/0.73 ((~![Y: $i, X: $i] : (complement(intersection(complement(X), complement(Y))) = union(X, Y))) | (complement(intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers)))) = union(singleton(null_class), image(successor_relation, ordinal_numbers)))),
% 0.71/0.73 inference(quant_inst,[status(thm)],[])).
% 0.71/0.73 tff(10,plain,
% 0.71/0.73 (complement(intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers)))) = union(singleton(null_class), image(successor_relation, ordinal_numbers))),
% 0.71/0.73 inference(unit_resolution,[status(thm)],[9, 8])).
% 0.71/0.73 tff(11,plain,
% 0.71/0.73 (member(null_class, complement(intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers))))) <=> member(null_class, union(singleton(null_class), image(successor_relation, ordinal_numbers)))),
% 0.71/0.73 inference(monotonicity,[status(thm)],[10])).
% 0.71/0.73 tff(12,plain,
% 0.71/0.73 (member(null_class, union(singleton(null_class), image(successor_relation, ordinal_numbers))) <=> member(null_class, complement(intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers)))))),
% 0.71/0.73 inference(symmetry,[status(thm)],[11])).
% 0.71/0.73 tff(13,plain,
% 0.71/0.73 ((~member(null_class, union(singleton(null_class), image(successor_relation, ordinal_numbers)))) <=> (~member(null_class, complement(intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers))))))),
% 0.71/0.73 inference(monotonicity,[status(thm)],[12])).
% 0.71/0.73 tff(14,plain,
% 0.71/0.73 (member(null_class, limit_ordinals) <=> member(null_class, intersection(complement(union(singleton(null_class), image(successor_relation, ordinal_numbers))), ordinal_numbers))),
% 0.71/0.73 inference(rewrite,[status(thm)],[])).
% 0.71/0.73 tff(15,plain,
% 0.71/0.73 (member(null_class, limit_ordinals) <=> member(null_class, limit_ordinals)),
% 0.71/0.73 inference(rewrite,[status(thm)],[])).
% 0.71/0.73 tff(16,axiom,(member(null_class, limit_ordinals)), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','prove_null_class_is_not_a_limit_ordinal_1')).
% 0.71/0.73 tff(17,plain,
% 0.71/0.73 (member(null_class, limit_ordinals)),
% 0.71/0.73 inference(modus_ponens,[status(thm)],[16, 15])).
% 0.71/0.73 tff(18,plain,
% 0.71/0.73 (member(null_class, intersection(complement(union(singleton(null_class), image(successor_relation, ordinal_numbers))), ordinal_numbers))),
% 0.71/0.73 inference(modus_ponens,[status(thm)],[17, 14])).
% 0.71/0.73 tff(19,plain,
% 0.71/0.73 (^[Z: $i, Y: $i, X: $i] : refl(((~member(Z, intersection(X, Y))) | member(Z, X)) <=> ((~member(Z, intersection(X, Y))) | member(Z, X)))),
% 0.71/0.73 inference(bind,[status(th)],[])).
% 0.71/0.73 tff(20,plain,
% 0.71/0.73 (![Z: $i, Y: $i, X: $i] : ((~member(Z, intersection(X, Y))) | member(Z, X)) <=> ![Z: $i, Y: $i, X: $i] : ((~member(Z, intersection(X, Y))) | member(Z, X))),
% 0.71/0.73 inference(quant_intro,[status(thm)],[19])).
% 0.71/0.73 tff(21,plain,
% 0.71/0.73 (![Z: $i, Y: $i, X: $i] : ((~member(Z, intersection(X, Y))) | member(Z, X)) <=> ![Z: $i, Y: $i, X: $i] : ((~member(Z, intersection(X, Y))) | member(Z, X))),
% 0.71/0.73 inference(rewrite,[status(thm)],[])).
% 0.71/0.73 tff(22,axiom,(![Z: $i, Y: $i, X: $i] : ((~member(Z, intersection(X, Y))) | member(Z, X))), file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax','intersection1')).
% 0.71/0.73 tff(23,plain,
% 0.71/0.73 (![Z: $i, Y: $i, X: $i] : ((~member(Z, intersection(X, Y))) | member(Z, X))),
% 0.71/0.73 inference(modus_ponens,[status(thm)],[22, 21])).
% 0.71/0.73 tff(24,plain,(
% 0.71/0.73 ![Z: $i, Y: $i, X: $i] : ((~member(Z, intersection(X, Y))) | member(Z, X))),
% 0.71/0.73 inference(skolemize,[status(sab)],[23])).
% 0.71/0.73 tff(25,plain,
% 0.71/0.73 (![Z: $i, Y: $i, X: $i] : ((~member(Z, intersection(X, Y))) | member(Z, X))),
% 0.71/0.73 inference(modus_ponens,[status(thm)],[24, 20])).
% 0.71/0.73 tff(26,plain,
% 0.71/0.73 (((~![Z: $i, Y: $i, X: $i] : ((~member(Z, intersection(X, Y))) | member(Z, X))) | ((~member(null_class, intersection(complement(union(singleton(null_class), image(successor_relation, ordinal_numbers))), ordinal_numbers))) | member(null_class, complement(union(singleton(null_class), image(successor_relation, ordinal_numbers)))))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~member(Z, intersection(X, Y))) | member(Z, X))) | (~member(null_class, intersection(complement(union(singleton(null_class), image(successor_relation, ordinal_numbers))), ordinal_numbers))) | member(null_class, complement(union(singleton(null_class), image(successor_relation, ordinal_numbers)))))),
% 0.71/0.74 inference(rewrite,[status(thm)],[])).
% 0.71/0.74 tff(27,plain,
% 0.71/0.74 ((~![Z: $i, Y: $i, X: $i] : ((~member(Z, intersection(X, Y))) | member(Z, X))) | ((~member(null_class, intersection(complement(union(singleton(null_class), image(successor_relation, ordinal_numbers))), ordinal_numbers))) | member(null_class, complement(union(singleton(null_class), image(successor_relation, ordinal_numbers)))))),
% 0.71/0.74 inference(quant_inst,[status(thm)],[])).
% 0.71/0.74 tff(28,plain,
% 0.71/0.74 ((~![Z: $i, Y: $i, X: $i] : ((~member(Z, intersection(X, Y))) | member(Z, X))) | (~member(null_class, intersection(complement(union(singleton(null_class), image(successor_relation, ordinal_numbers))), ordinal_numbers))) | member(null_class, complement(union(singleton(null_class), image(successor_relation, ordinal_numbers))))),
% 0.71/0.74 inference(modus_ponens,[status(thm)],[27, 26])).
% 0.71/0.74 tff(29,plain,
% 0.71/0.74 (member(null_class, complement(union(singleton(null_class), image(successor_relation, ordinal_numbers))))),
% 0.71/0.74 inference(unit_resolution,[status(thm)],[28, 25, 18])).
% 0.71/0.74 tff(30,plain,
% 0.71/0.74 (^[Z: $i, X: $i] : refl(((~member(Z, X)) | (~member(Z, complement(X)))) <=> ((~member(Z, X)) | (~member(Z, complement(X)))))),
% 0.71/0.74 inference(bind,[status(th)],[])).
% 0.71/0.74 tff(31,plain,
% 0.71/0.74 (![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X)))) <=> ![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))),
% 0.71/0.74 inference(quant_intro,[status(thm)],[30])).
% 0.71/0.74 tff(32,plain,
% 0.71/0.74 (![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X)))) <=> ![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))),
% 0.71/0.74 inference(rewrite,[status(thm)],[])).
% 0.71/0.74 tff(33,plain,
% 0.81/0.74 (^[Z: $i, X: $i] : rewrite(((~member(Z, complement(X))) | (~member(Z, X))) <=> ((~member(Z, X)) | (~member(Z, complement(X)))))),
% 0.81/0.74 inference(bind,[status(th)],[])).
% 0.81/0.74 tff(34,plain,
% 0.81/0.74 (![Z: $i, X: $i] : ((~member(Z, complement(X))) | (~member(Z, X))) <=> ![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))),
% 0.81/0.74 inference(quant_intro,[status(thm)],[33])).
% 0.81/0.74 tff(35,axiom,(![Z: $i, X: $i] : ((~member(Z, complement(X))) | (~member(Z, X)))), file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax','complement1')).
% 0.81/0.74 tff(36,plain,
% 0.81/0.74 (![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))),
% 0.81/0.74 inference(modus_ponens,[status(thm)],[35, 34])).
% 0.81/0.74 tff(37,plain,
% 0.81/0.74 (![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))),
% 0.81/0.74 inference(modus_ponens,[status(thm)],[36, 32])).
% 0.81/0.74 tff(38,plain,(
% 0.81/0.74 ![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))),
% 0.81/0.74 inference(skolemize,[status(sab)],[37])).
% 0.81/0.74 tff(39,plain,
% 0.81/0.74 (![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))),
% 0.81/0.74 inference(modus_ponens,[status(thm)],[38, 31])).
% 0.81/0.74 tff(40,plain,
% 0.81/0.74 (((~![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))) | ((~member(null_class, union(singleton(null_class), image(successor_relation, ordinal_numbers)))) | (~member(null_class, complement(union(singleton(null_class), image(successor_relation, ordinal_numbers))))))) <=> ((~![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))) | (~member(null_class, union(singleton(null_class), image(successor_relation, ordinal_numbers)))) | (~member(null_class, complement(union(singleton(null_class), image(successor_relation, ordinal_numbers))))))),
% 0.81/0.74 inference(rewrite,[status(thm)],[])).
% 0.81/0.74 tff(41,plain,
% 0.81/0.74 ((~![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))) | ((~member(null_class, union(singleton(null_class), image(successor_relation, ordinal_numbers)))) | (~member(null_class, complement(union(singleton(null_class), image(successor_relation, ordinal_numbers))))))),
% 0.81/0.74 inference(quant_inst,[status(thm)],[])).
% 0.81/0.74 tff(42,plain,
% 0.81/0.74 ((~![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))) | (~member(null_class, union(singleton(null_class), image(successor_relation, ordinal_numbers)))) | (~member(null_class, complement(union(singleton(null_class), image(successor_relation, ordinal_numbers)))))),
% 0.81/0.74 inference(modus_ponens,[status(thm)],[41, 40])).
% 0.81/0.74 tff(43,plain,
% 0.81/0.74 (~member(null_class, union(singleton(null_class), image(successor_relation, ordinal_numbers)))),
% 0.81/0.74 inference(unit_resolution,[status(thm)],[42, 39, 29])).
% 0.81/0.74 tff(44,plain,
% 0.81/0.74 (~member(null_class, complement(intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers)))))),
% 0.81/0.74 inference(modus_ponens,[status(thm)],[43, 13])).
% 0.81/0.74 tff(45,plain,
% 0.81/0.74 (^[Z: $i, X: $i] : refl((member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X))) <=> (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X))))),
% 0.81/0.74 inference(bind,[status(th)],[])).
% 0.81/0.74 tff(46,plain,
% 0.81/0.74 (![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X))) <=> ![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))),
% 0.81/0.74 inference(quant_intro,[status(thm)],[45])).
% 0.81/0.74 tff(47,plain,
% 0.81/0.74 (![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X))) <=> ![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))),
% 0.81/0.74 inference(rewrite,[status(thm)],[])).
% 0.81/0.74 tff(48,plain,
% 0.81/0.74 (^[Z: $i, X: $i] : rewrite((((~member(Z, universal_class)) | member(Z, complement(X))) | member(Z, X)) <=> (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X))))),
% 0.81/0.74 inference(bind,[status(th)],[])).
% 0.81/0.74 tff(49,plain,
% 0.81/0.74 (![Z: $i, X: $i] : (((~member(Z, universal_class)) | member(Z, complement(X))) | member(Z, X)) <=> ![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))),
% 0.81/0.74 inference(quant_intro,[status(thm)],[48])).
% 0.81/0.74 tff(50,axiom,(![Z: $i, X: $i] : (((~member(Z, universal_class)) | member(Z, complement(X))) | member(Z, X))), file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax','complement2')).
% 0.81/0.74 tff(51,plain,
% 0.81/0.74 (![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))),
% 0.81/0.74 inference(modus_ponens,[status(thm)],[50, 49])).
% 0.81/0.74 tff(52,plain,
% 0.81/0.74 (![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))),
% 0.81/0.74 inference(modus_ponens,[status(thm)],[51, 47])).
% 0.81/0.74 tff(53,plain,(
% 0.81/0.74 ![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))),
% 0.81/0.74 inference(skolemize,[status(sab)],[52])).
% 0.81/0.74 tff(54,plain,
% 0.81/0.74 (![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))),
% 0.81/0.74 inference(modus_ponens,[status(thm)],[53, 46])).
% 0.81/0.74 tff(55,plain,
% 0.81/0.74 (((~![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))) | ((~member(null_class, universal_class)) | member(null_class, intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers)))) | member(null_class, complement(intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers))))))) <=> ((~![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))) | (~member(null_class, universal_class)) | member(null_class, intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers)))) | member(null_class, complement(intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers))))))),
% 0.81/0.74 inference(rewrite,[status(thm)],[])).
% 0.81/0.74 tff(56,plain,
% 0.81/0.74 ((member(null_class, intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers)))) | (~member(null_class, universal_class)) | member(null_class, complement(intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers)))))) <=> ((~member(null_class, universal_class)) | member(null_class, intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers)))) | member(null_class, complement(intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers))))))),
% 0.81/0.74 inference(rewrite,[status(thm)],[])).
% 0.81/0.74 tff(57,plain,
% 0.81/0.74 (((~![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))) | (member(null_class, intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers)))) | (~member(null_class, universal_class)) | member(null_class, complement(intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers))))))) <=> ((~![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))) | ((~member(null_class, universal_class)) | member(null_class, intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers)))) | member(null_class, complement(intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers)))))))),
% 0.81/0.74 inference(monotonicity,[status(thm)],[56])).
% 0.81/0.74 tff(58,plain,
% 0.81/0.74 (((~![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))) | (member(null_class, intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers)))) | (~member(null_class, universal_class)) | member(null_class, complement(intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers))))))) <=> ((~![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))) | (~member(null_class, universal_class)) | member(null_class, intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers)))) | member(null_class, complement(intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers))))))),
% 0.81/0.74 inference(transitivity,[status(thm)],[57, 55])).
% 0.81/0.74 tff(59,plain,
% 0.81/0.74 ((~![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))) | (member(null_class, intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers)))) | (~member(null_class, universal_class)) | member(null_class, complement(intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers))))))),
% 0.81/0.74 inference(quant_inst,[status(thm)],[])).
% 0.81/0.74 tff(60,plain,
% 0.81/0.74 ((~![Z: $i, X: $i] : (member(Z, X) | (~member(Z, universal_class)) | member(Z, complement(X)))) | (~member(null_class, universal_class)) | member(null_class, intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers)))) | member(null_class, complement(intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers)))))),
% 0.81/0.74 inference(modus_ponens,[status(thm)],[59, 58])).
% 0.81/0.74 tff(61,plain,
% 0.81/0.74 ((~member(null_class, universal_class)) | member(null_class, intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers)))) | member(null_class, complement(intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers)))))),
% 0.81/0.74 inference(unit_resolution,[status(thm)],[60, 54])).
% 0.81/0.74 tff(62,plain,
% 0.81/0.74 ((~member(null_class, universal_class)) | member(null_class, intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers))))),
% 0.81/0.74 inference(unit_resolution,[status(thm)],[61, 44])).
% 0.81/0.74 tff(63,plain,
% 0.81/0.74 (member(null_class, intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers))))),
% 0.81/0.74 inference(unit_resolution,[status(thm)],[62, 1])).
% 0.81/0.74 tff(64,plain,
% 0.81/0.74 (((~![Z: $i, Y: $i, X: $i] : ((~member(Z, intersection(X, Y))) | member(Z, X))) | ((~member(null_class, intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers))))) | member(null_class, complement(singleton(null_class))))) <=> ((~![Z: $i, Y: $i, X: $i] : ((~member(Z, intersection(X, Y))) | member(Z, X))) | (~member(null_class, intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers))))) | member(null_class, complement(singleton(null_class))))),
% 0.81/0.74 inference(rewrite,[status(thm)],[])).
% 0.81/0.74 tff(65,plain,
% 0.81/0.74 ((~![Z: $i, Y: $i, X: $i] : ((~member(Z, intersection(X, Y))) | member(Z, X))) | ((~member(null_class, intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers))))) | member(null_class, complement(singleton(null_class))))),
% 0.81/0.74 inference(quant_inst,[status(thm)],[])).
% 0.81/0.74 tff(66,plain,
% 0.81/0.74 ((~![Z: $i, Y: $i, X: $i] : ((~member(Z, intersection(X, Y))) | member(Z, X))) | (~member(null_class, intersection(complement(singleton(null_class)), complement(image(successor_relation, ordinal_numbers))))) | member(null_class, complement(singleton(null_class)))),
% 0.81/0.74 inference(modus_ponens,[status(thm)],[65, 64])).
% 0.81/0.74 tff(67,plain,
% 0.81/0.74 (member(null_class, complement(singleton(null_class)))),
% 0.81/0.74 inference(unit_resolution,[status(thm)],[66, 25, 63])).
% 0.81/0.74 tff(68,plain,
% 0.81/0.74 (^[X: $i] : refl((unordered_pair(X, X) = singleton(X)) <=> (unordered_pair(X, X) = singleton(X)))),
% 0.81/0.74 inference(bind,[status(th)],[])).
% 0.81/0.74 tff(69,plain,
% 0.81/0.74 (![X: $i] : (unordered_pair(X, X) = singleton(X)) <=> ![X: $i] : (unordered_pair(X, X) = singleton(X))),
% 0.81/0.74 inference(quant_intro,[status(thm)],[68])).
% 0.81/0.74 tff(70,plain,
% 0.81/0.74 (![X: $i] : (unordered_pair(X, X) = singleton(X)) <=> ![X: $i] : (unordered_pair(X, X) = singleton(X))),
% 0.81/0.74 inference(rewrite,[status(thm)],[])).
% 0.81/0.74 tff(71,axiom,(![X: $i] : (unordered_pair(X, X) = singleton(X))), file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax','singleton_set')).
% 0.81/0.74 tff(72,plain,
% 0.81/0.74 (![X: $i] : (unordered_pair(X, X) = singleton(X))),
% 0.81/0.74 inference(modus_ponens,[status(thm)],[71, 70])).
% 0.81/0.74 tff(73,plain,(
% 0.81/0.74 ![X: $i] : (unordered_pair(X, X) = singleton(X))),
% 0.81/0.74 inference(skolemize,[status(sab)],[72])).
% 0.81/0.74 tff(74,plain,
% 0.81/0.74 (![X: $i] : (unordered_pair(X, X) = singleton(X))),
% 0.81/0.74 inference(modus_ponens,[status(thm)],[73, 69])).
% 0.81/0.74 tff(75,plain,
% 0.81/0.74 ((~![X: $i] : (unordered_pair(X, X) = singleton(X))) | (unordered_pair(null_class, null_class) = singleton(null_class))),
% 0.81/0.74 inference(quant_inst,[status(thm)],[])).
% 0.81/0.74 tff(76,plain,
% 0.81/0.74 (unordered_pair(null_class, null_class) = singleton(null_class)),
% 0.81/0.74 inference(unit_resolution,[status(thm)],[75, 74])).
% 0.81/0.74 tff(77,plain,
% 0.81/0.74 (singleton(null_class) = unordered_pair(null_class, null_class)),
% 0.81/0.74 inference(symmetry,[status(thm)],[76])).
% 0.81/0.74 tff(78,plain,
% 0.81/0.74 (member(null_class, singleton(null_class)) <=> member(null_class, unordered_pair(null_class, null_class))),
% 0.81/0.74 inference(monotonicity,[status(thm)],[77])).
% 0.81/0.74 tff(79,plain,
% 0.81/0.74 (member(null_class, unordered_pair(null_class, null_class)) <=> member(null_class, singleton(null_class))),
% 0.81/0.74 inference(symmetry,[status(thm)],[78])).
% 0.81/0.74 tff(80,plain,
% 0.81/0.74 (^[Y: $i, X: $i] : refl(((~member(X, universal_class)) | member(X, unordered_pair(X, Y))) <=> ((~member(X, universal_class)) | member(X, unordered_pair(X, Y))))),
% 0.81/0.74 inference(bind,[status(th)],[])).
% 0.81/0.74 tff(81,plain,
% 0.81/0.74 (![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y))) <=> ![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))),
% 0.81/0.74 inference(quant_intro,[status(thm)],[80])).
% 0.81/0.74 tff(82,plain,
% 0.81/0.74 (![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y))) <=> ![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))),
% 0.81/0.74 inference(rewrite,[status(thm)],[])).
% 0.81/0.74 tff(83,axiom,(![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))), file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax','unordered_pair2')).
% 0.81/0.74 tff(84,plain,
% 0.81/0.74 (![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))),
% 0.81/0.74 inference(modus_ponens,[status(thm)],[83, 82])).
% 0.81/0.74 tff(85,plain,(
% 0.81/0.74 ![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))),
% 0.81/0.74 inference(skolemize,[status(sab)],[84])).
% 0.81/0.74 tff(86,plain,
% 0.81/0.74 (![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))),
% 0.81/0.74 inference(modus_ponens,[status(thm)],[85, 81])).
% 0.81/0.74 tff(87,plain,
% 0.81/0.74 (((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | ((~member(null_class, universal_class)) | member(null_class, unordered_pair(null_class, null_class)))) <=> ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | (~member(null_class, universal_class)) | member(null_class, unordered_pair(null_class, null_class)))),
% 0.81/0.74 inference(rewrite,[status(thm)],[])).
% 0.81/0.74 tff(88,plain,
% 0.81/0.74 ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | ((~member(null_class, universal_class)) | member(null_class, unordered_pair(null_class, null_class)))),
% 0.81/0.74 inference(quant_inst,[status(thm)],[])).
% 0.81/0.74 tff(89,plain,
% 0.81/0.74 ((~![Y: $i, X: $i] : ((~member(X, universal_class)) | member(X, unordered_pair(X, Y)))) | (~member(null_class, universal_class)) | member(null_class, unordered_pair(null_class, null_class))),
% 0.81/0.74 inference(modus_ponens,[status(thm)],[88, 87])).
% 0.81/0.74 tff(90,plain,
% 0.81/0.74 ((~member(null_class, universal_class)) | member(null_class, unordered_pair(null_class, null_class))),
% 0.81/0.74 inference(unit_resolution,[status(thm)],[89, 86])).
% 0.81/0.74 tff(91,plain,
% 0.81/0.74 (member(null_class, unordered_pair(null_class, null_class))),
% 0.81/0.74 inference(unit_resolution,[status(thm)],[90, 1])).
% 0.81/0.74 tff(92,plain,
% 0.81/0.74 (member(null_class, singleton(null_class))),
% 0.81/0.74 inference(modus_ponens,[status(thm)],[91, 79])).
% 0.81/0.74 tff(93,plain,
% 0.81/0.74 (((~![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))) | ((~member(null_class, singleton(null_class))) | (~member(null_class, complement(singleton(null_class)))))) <=> ((~![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))) | (~member(null_class, singleton(null_class))) | (~member(null_class, complement(singleton(null_class)))))),
% 0.81/0.74 inference(rewrite,[status(thm)],[])).
% 0.81/0.74 tff(94,plain,
% 0.81/0.74 ((~![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))) | ((~member(null_class, singleton(null_class))) | (~member(null_class, complement(singleton(null_class)))))),
% 0.81/0.74 inference(quant_inst,[status(thm)],[])).
% 0.81/0.74 tff(95,plain,
% 0.81/0.74 ((~![Z: $i, X: $i] : ((~member(Z, X)) | (~member(Z, complement(X))))) | (~member(null_class, singleton(null_class))) | (~member(null_class, complement(singleton(null_class))))),
% 0.81/0.74 inference(modus_ponens,[status(thm)],[94, 93])).
% 0.81/0.74 tff(96,plain,
% 0.81/0.74 ($false),
% 0.81/0.74 inference(unit_resolution,[status(thm)],[95, 39, 92, 67])).
% 0.81/0.74 tff(97,plain,(~member(null_class, universal_class)), inference(lemma,lemma(discharge,[]))).
% 0.81/0.74 tff(98,plain,
% 0.81/0.74 (^[Y: $i, U: $i, X: $i] : refl((member(U, Y) | (~member(U, X)) | (~subclass(X, Y))) <=> (member(U, Y) | (~member(U, X)) | (~subclass(X, Y))))),
% 0.81/0.74 inference(bind,[status(th)],[])).
% 0.81/0.74 tff(99,plain,
% 0.81/0.74 (![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y))) <=> ![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))),
% 0.81/0.74 inference(quant_intro,[status(thm)],[98])).
% 0.81/0.74 tff(100,plain,
% 0.81/0.74 (![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y))) <=> ![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))),
% 0.81/0.74 inference(rewrite,[status(thm)],[])).
% 0.81/0.74 tff(101,plain,
% 0.81/0.74 (^[Y: $i, U: $i, X: $i] : trans(monotonicity(rewrite(((~subclass(X, Y)) | (~member(U, X))) <=> ((~member(U, X)) | (~subclass(X, Y)))), ((((~subclass(X, Y)) | (~member(U, X))) | member(U, Y)) <=> (((~member(U, X)) | (~subclass(X, Y))) | member(U, Y)))), rewrite((((~member(U, X)) | (~subclass(X, Y))) | member(U, Y)) <=> (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))), ((((~subclass(X, Y)) | (~member(U, X))) | member(U, Y)) <=> (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))))),
% 0.81/0.74 inference(bind,[status(th)],[])).
% 0.81/0.74 tff(102,plain,
% 0.81/0.74 (![Y: $i, U: $i, X: $i] : (((~subclass(X, Y)) | (~member(U, X))) | member(U, Y)) <=> ![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))),
% 0.81/0.74 inference(quant_intro,[status(thm)],[101])).
% 0.81/0.74 tff(103,axiom,(![Y: $i, U: $i, X: $i] : (((~subclass(X, Y)) | (~member(U, X))) | member(U, Y))), file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax','subclass_members')).
% 0.81/0.74 tff(104,plain,
% 0.81/0.74 (![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))),
% 0.81/0.74 inference(modus_ponens,[status(thm)],[103, 102])).
% 0.81/0.74 tff(105,plain,
% 0.81/0.74 (![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))),
% 0.81/0.74 inference(modus_ponens,[status(thm)],[104, 100])).
% 0.81/0.74 tff(106,plain,(
% 0.81/0.74 ![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))),
% 0.81/0.74 inference(skolemize,[status(sab)],[105])).
% 0.81/0.74 tff(107,plain,
% 0.81/0.74 (![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))),
% 0.81/0.74 inference(modus_ponens,[status(thm)],[106, 99])).
% 0.81/0.74 tff(108,plain,
% 0.81/0.74 (((~![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))) | ((~member(null_class, intersection(complement(union(singleton(null_class), image(successor_relation, ordinal_numbers))), ordinal_numbers))) | member(null_class, universal_class) | (~subclass(intersection(complement(union(singleton(null_class), image(successor_relation, ordinal_numbers))), ordinal_numbers), universal_class)))) <=> ((~![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))) | (~member(null_class, intersection(complement(union(singleton(null_class), image(successor_relation, ordinal_numbers))), ordinal_numbers))) | member(null_class, universal_class) | (~subclass(intersection(complement(union(singleton(null_class), image(successor_relation, ordinal_numbers))), ordinal_numbers), universal_class)))),
% 0.81/0.74 inference(rewrite,[status(thm)],[])).
% 0.81/0.74 tff(109,plain,
% 0.81/0.74 ((member(null_class, universal_class) | (~member(null_class, intersection(complement(union(singleton(null_class), image(successor_relation, ordinal_numbers))), ordinal_numbers))) | (~subclass(intersection(complement(union(singleton(null_class), image(successor_relation, ordinal_numbers))), ordinal_numbers), universal_class))) <=> ((~member(null_class, intersection(complement(union(singleton(null_class), image(successor_relation, ordinal_numbers))), ordinal_numbers))) | member(null_class, universal_class) | (~subclass(intersection(complement(union(singleton(null_class), image(successor_relation, ordinal_numbers))), ordinal_numbers), universal_class)))),
% 0.81/0.74 inference(rewrite,[status(thm)],[])).
% 0.81/0.74 tff(110,plain,
% 0.81/0.74 (((~![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))) | (member(null_class, universal_class) | (~member(null_class, intersection(complement(union(singleton(null_class), image(successor_relation, ordinal_numbers))), ordinal_numbers))) | (~subclass(intersection(complement(union(singleton(null_class), image(successor_relation, ordinal_numbers))), ordinal_numbers), universal_class)))) <=> ((~![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))) | ((~member(null_class, intersection(complement(union(singleton(null_class), image(successor_relation, ordinal_numbers))), ordinal_numbers))) | member(null_class, universal_class) | (~subclass(intersection(complement(union(singleton(null_class), image(successor_relation, ordinal_numbers))), ordinal_numbers), universal_class))))),
% 0.81/0.74 inference(monotonicity,[status(thm)],[109])).
% 0.81/0.74 tff(111,plain,
% 0.81/0.74 (((~![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))) | (member(null_class, universal_class) | (~member(null_class, intersection(complement(union(singleton(null_class), image(successor_relation, ordinal_numbers))), ordinal_numbers))) | (~subclass(intersection(complement(union(singleton(null_class), image(successor_relation, ordinal_numbers))), ordinal_numbers), universal_class)))) <=> ((~![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))) | (~member(null_class, intersection(complement(union(singleton(null_class), image(successor_relation, ordinal_numbers))), ordinal_numbers))) | member(null_class, universal_class) | (~subclass(intersection(complement(union(singleton(null_class), image(successor_relation, ordinal_numbers))), ordinal_numbers), universal_class)))),
% 0.81/0.74 inference(transitivity,[status(thm)],[110, 108])).
% 0.81/0.74 tff(112,plain,
% 0.81/0.74 ((~![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))) | (member(null_class, universal_class) | (~member(null_class, intersection(complement(union(singleton(null_class), image(successor_relation, ordinal_numbers))), ordinal_numbers))) | (~subclass(intersection(complement(union(singleton(null_class), image(successor_relation, ordinal_numbers))), ordinal_numbers), universal_class)))),
% 0.81/0.74 inference(quant_inst,[status(thm)],[])).
% 0.81/0.74 tff(113,plain,
% 0.81/0.74 ((~![Y: $i, U: $i, X: $i] : (member(U, Y) | (~member(U, X)) | (~subclass(X, Y)))) | (~member(null_class, intersection(complement(union(singleton(null_class), image(successor_relation, ordinal_numbers))), ordinal_numbers))) | member(null_class, universal_class) | (~subclass(intersection(complement(union(singleton(null_class), image(successor_relation, ordinal_numbers))), ordinal_numbers), universal_class))),
% 0.81/0.74 inference(modus_ponens,[status(thm)],[112, 111])).
% 0.81/0.74 tff(114,plain,
% 0.81/0.74 (~subclass(intersection(complement(union(singleton(null_class), image(successor_relation, ordinal_numbers))), ordinal_numbers), universal_class)),
% 0.81/0.74 inference(unit_resolution,[status(thm)],[113, 107, 18, 97])).
% 0.81/0.74 tff(115,plain,
% 0.81/0.74 (^[X: $i] : refl(subclass(X, universal_class) <=> subclass(X, universal_class))),
% 0.81/0.74 inference(bind,[status(th)],[])).
% 0.81/0.74 tff(116,plain,
% 0.81/0.74 (![X: $i] : subclass(X, universal_class) <=> ![X: $i] : subclass(X, universal_class)),
% 0.81/0.74 inference(quant_intro,[status(thm)],[115])).
% 0.81/0.74 tff(117,plain,
% 0.81/0.74 (![X: $i] : subclass(X, universal_class) <=> ![X: $i] : subclass(X, universal_class)),
% 0.81/0.74 inference(rewrite,[status(thm)],[])).
% 0.81/0.74 tff(118,axiom,(![X: $i] : subclass(X, universal_class)), file('/export/starexec/sandbox2/benchmark/Axioms/SET004-0.ax','class_elements_are_sets')).
% 0.81/0.74 tff(119,plain,
% 0.81/0.74 (![X: $i] : subclass(X, universal_class)),
% 0.81/0.74 inference(modus_ponens,[status(thm)],[118, 117])).
% 0.81/0.74 tff(120,plain,(
% 0.81/0.75 ![X: $i] : subclass(X, universal_class)),
% 0.81/0.75 inference(skolemize,[status(sab)],[119])).
% 0.81/0.75 tff(121,plain,
% 0.81/0.75 (![X: $i] : subclass(X, universal_class)),
% 0.81/0.75 inference(modus_ponens,[status(thm)],[120, 116])).
% 0.81/0.75 tff(122,plain,
% 0.81/0.75 ((~![X: $i] : subclass(X, universal_class)) | subclass(intersection(complement(union(singleton(null_class), image(successor_relation, ordinal_numbers))), ordinal_numbers), universal_class)),
% 0.81/0.75 inference(quant_inst,[status(thm)],[])).
% 0.81/0.75 tff(123,plain,
% 0.81/0.75 ($false),
% 0.81/0.75 inference(unit_resolution,[status(thm)],[122, 121, 114])).
% 0.81/0.75 % SZS output end Proof
%------------------------------------------------------------------------------